Optimal. Leaf size=40 \[ \frac{2 \sqrt{x}}{a}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2}} \]
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Rubi [A] time = 0.013709, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {263, 50, 63, 205} \[ \frac{2 \sqrt{x}}{a}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 263
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right ) \sqrt{x}} \, dx &=\int \frac{\sqrt{x}}{b+a x} \, dx\\ &=\frac{2 \sqrt{x}}{a}-\frac{b \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{a}\\ &=\frac{2 \sqrt{x}}{a}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{2 \sqrt{x}}{a}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0093032, size = 40, normalized size = 1. \[ \frac{2 \sqrt{x}}{a}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 32, normalized size = 0.8 \begin{align*} 2\,{\frac{\sqrt{x}}{a}}-2\,{\frac{b}{a\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80438, size = 189, normalized size = 4.72 \begin{align*} \left [\frac{\sqrt{-\frac{b}{a}} \log \left (\frac{a x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \, \sqrt{x}}{a}, -\frac{2 \,{\left (\sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{x} \sqrt{\frac{b}{a}}}{b}\right ) - \sqrt{x}\right )}}{a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.37189, size = 92, normalized size = 2.3 \begin{align*} \begin{cases} \frac{2 \sqrt{x}}{a} + \frac{i \sqrt{b} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{2} \sqrt{\frac{1}{a}}} - \frac{i \sqrt{b} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{2} \sqrt{\frac{1}{a}}} & \text{for}\: a \neq 0 \\\frac{2 x^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08196, size = 42, normalized size = 1.05 \begin{align*} -\frac{2 \, b \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a} + \frac{2 \, \sqrt{x}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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